Simple and Compound Interest and Yield

# Interest

Interest is the amount paid (or charged) for the use of money.  A borrower must pay for being able to use someone elses money.  A lender is paid for letting someone else use her/his/its money.   The amount of interest to pay or be paid can be calculated on either a simple or a compound basis.

# Simple Interest

Simple interest is calculated by considering only the outstanding principal (the amount borrowed or lent and not yet repaid).  The equation for doing this is I = P x R x T.  Where I = Interest, P = Principal outstanding, R = Rate of interest per year, and T = Time as a portion or fraction of a year.  Note: In determining the length of time a loan or investment is outstanding it is common practice to count the day the loan (investment) is made but not to count the day it is paid off or cashed out.  So, for example, the length of a loan made on October 28, 20x1 and paid off on November 14, 20x1 would be 17 days.  The last four days of October and the first 13 days of November.

As an example assume Sandy Turcott borrows \$10,000, which is due in two months with 8% interest.  After two months Sandy must repay the \$10,000 principal and 8% interest of \$133 (\$10,000 x .08 x 2/12).  If it were a two-year loan then the interest would be \$1,600 (\$10,000 x .08 x 2).  Notice there is no interest in the second year on the interest that was earned in the first year.  This is not normally used in business because the \$800 interest due after the first year is really equivalent to another \$800 one-year loan that is being given interest free.

# Compound Interest

Compound interest is calculated on the outstanding principal and the previously calculated but unpaid or uncollected interest.  To do this one needs to know both the compounding period and the number of those compounding periods in the life of the loan.  To determine the amount of the payment necessary to repay the loan and its associated interest one uses the equation FV = P x (1+i)ⁿ where FV = Future Value, P = Principal amount of the loan, i = interest rate per compounding period stated as a decimal and n = the number of compounding periods in the life of the loan.  Note: The interest rate must match the time period.  For example if interest is compounded every quarter then one must use a quarterly interest rate and if the interest is compounded every year then one must use a yearly interest rate.  Also note that if there is only one compounding period then compound interest and simple interest are the same.

In the example from the preceding paragraph with the two-month loan there is one two-month compounding period and the interest rate per two months is 1.33% (8% x 2/12) or .0133.  Using the above formula we get FV = \$10,000 x (1.0133) so FV = \$10,133.  The amount includes both the principal and the interest.  The principal is \$10,000; therefore, the interest must be \$133.  For the two-year example there are two one-year compounding periods and the interest rate per year is 8%.  Using the compound interest equation we get FV = \$10,000 x (1.08) = \$10,000 x 1.1664 = \$11,664.  After subtracting the principal we see that the interest for the two-year period is \$1,664.  (If one wanted to directly solve for the amount of interest the formula is I = (P x ((1+i)ⁿ-1)).   The compound interest is \$64 more than the simple interest for the same period because compound interest computes interest on interest.  The \$800 interest of the first period earns 8% interest in the second period and \$800 x .08 is \$64.

To reinforce the applications of the equations (formulas) introduced above solve the following problems:

How much will it cost to borrow \$500 for three months if the going rate of interest is 9% per annum compounded quarterly?  This can be determined using either the simple interest or compound interest formula because with only one compounding period the results of the two are the same.  Using I = P x I x T we get I = \$500 x .09 x 90/365 = \$11.10.   Using I = (P x ((1+i)ⁿ - 1)) we get I = \$500 x ((1.022192)) - 1 = \$500 x .022192 = \$11.10.

You need to borrow \$15,000 for three years.  A financial institution agrees to lend it to you at 8.5% interest compounded annually with the principal and accrued interest due in three years at the maturity of the note.  What is the total amount that you must pay at that time?  How much of the payment is interest?  To solve this one needs to determine the future value of the \$15,000 principal and its accumulated interest.  FV = P x (1+i)ⁿ so FV = \$15,000 x (1.085)  = \$15,000 x 1.277289 = \$19,159.34.  To determine the amount of the interest one can just subtract the principal amount of \$15,000 from the total amount of \$19,159.34 and get \$4,159.34 or use the formula I = (P x ((1+i)ⁿ) 1)) which gives I = \$15,000 x ((1.085)- 1) = \$15,000 x .277289 = \$4,159.34.

In practice a financial institution is not likely to have you pay all the interest at the maturity of the loan unless its duration is quite short, 30 to 45 days or so. Instead it will require you to pay interest each month.  When this is the case the financial institution must determine an effective daily interest rate and charge you according to the number of days since you last paid interest.  As was stated earlier the convention in practice is to charge interest for the day you borrow money but not for the day you repay it.  Let us now assume that the \$15,000 three-year loan at 8.5% interest per annum compounded annually requires interest to be paid on the 5th of each month.  The loan was originally made on the 5th of September.  By the 5th of October 30 days have passed.  The effective daily interest rate is .00022353.  The interest due on October 5th is \$15,000 x .00022353 x 30 = \$100.59.

Notice this is just the application of the simple interest equation of I = P x R x T where rate and time are stated on a daily basis rather than a yearly one.  The complication is in determining the effective daily interest rate.  Because the interest rate of the loan is 8.5% per annum compounded annually then for a daily basis one needs to find out what rate, compounded daily, is equivalent to 8.5% compounded yearly.  This is done by using the compound interest equation I = (P x ((1+i)ⁿ)  1) and setting P = 1 and n = 1/365.  Setting P = 1 effectively changes the I from an absolute dollar amount to a decimal and n is set at 1/365 because that is the portion of a year made up by one day.  Doing the calculation then one gets I = (1 x ((1 + .085))  1) = 1 x .00022353 = .00022353 which is the effective daily interest rate.

Yield

The yield on an investment relates the amount earned from the investment to the amount of the investment.  It is common practice to state this amount as an annual percentage rate.  This is done by comparing the yearly earnings of the investment to the amount invested for a year.  By having all yields or returns stated on an annual basis it makes it easier to compare the return on one investment to that of another.

There are two ways to calculate yield.  One gives an approximation of the effective return and is often used when precision or absolute accuracy is not necessary.  This rate is often referred to as the annual percentage rate (APR), nominal rate or stated rate.  It is calculated by APR = (amount earned / amount invested) x (365/ # of days of the investment).  If one borrowed money rather than invested it the formula would be APR = (interest paid / amount borrowed) x (365/ # of days of the loan).   If one earns \$100 from having \$1,000 invested for one year then the APR is (100/1,000) x (365/365) = .1 X 1 = .10 or 10%.  If one earns \$50 for having \$1,000 invested for 182.5 days the yield (APR) is (50/1,000) x 365/182.5) = .05 x 2 = .10 or 10%.

The second way to calculate yield gives the effective yield.  This is the one that needs to be used when rate comparisons are being made and when future or present value calculations are being done.  The effective yield takes into consideration the effects of compound interest during a year.  It assumes the returns on an investment of less than a year can be duplicated for the rest of the year.  Its calculation is similar to that of the APR except it handles the effects of time differently.  It is computed by ((1+ amt. earned / amt. invested)) 1.  If you borrow rather than invest it is ((1+interest paid / amount borrowed)365 / # of days of the loan)  1.  If the investment is held for one year the APR and the effective yield are the same.  It is when the length of the investment is for other than a year that there is a difference between the APR and the effective rate.

For \$100 earned on  \$1,000 invested for one year we get ((1+ 100/1,000)) 1 or (1 + .1))  1 which equals 1.1  1 = .1.  This as you see comes up with the same answer as the APR calculation because it is for one years time.

For \$50 earned on \$1,000 invested for 182.5 days we get ((1+50/1,000))  1 or (1= .05) - 1 = 1.1025 1 = .1025 or 10. 25%.  Using the APR we got .10 or 10%.  Its slightly less (a .25% difference) but not a whole lot different.  The difference between the APR and the effective rate increases as the number of times of compounding per year increases.

If we look at a \$25 return for a \$1,000 investment for 91.25 days we get an APR of 25/1,000 x 365/91.25 = .025 x 4 = .10 or 10%.  We get an effective rate of  ((1+.025) -1 = .103813 or 10.3813%.  The difference is now .3813%.

Practice exercises 

1. D.J. Beardy has \$8,000 to invest.  She decides to lend it to a local business, which needs funds to increase its inventory.  They agree upon a loan of the total amount to be made on September 15th and to be repaid with interest at 7% per annum on November 30th.

Required: (a) Using the simple interest formula determine the interest Ms. Beardy will earn.  (b) Do the same thing using the compound interest formula.

2. James Chan borrowed \$570 and had to pay \$600 three months later to repay the loan.  What was the APR of interest he was charged?  What was the effective annual rate of interest he was charged?

3. How much will Eileen McTavish have to repay if she borrows \$5,000 for a year and a half when the interest rate is 8.5% per annum and all interest and principal are paid upon maturity?

4. If you buy a 90-day money market instrument for \$9,780 hold it for 46 days then sell it for \$9,890 what yield have you earned?

5.  What is the effective daily interest rate of interest at 8% compounded semi-annually?

See next page for solutions

1.  (a) Remember the simple interest formula is I = P X R x T.  In this instance we know that P = \$8,000, R = 7% or as a decimal .07 and we can determine T to be 76 days (16 days in September, 31 days in October, and 29 days in November  you count the day the loan is made but not the day it is paid).  This leads to I = \$8,000 x .07 x 76/365 which then becomes I = \$8,000 x .07 x .208219 finally solving for I we get \$116.60.

(b) The compound interest formula is FV = P x (1+i)n.  Using the information developed above we can determine that the interest rate for the 76 day period is .01457533 (.07 x .208219) we then have FV = \$8,000 x (1.01457533)1.  This then becomes FV = \$8,000 x (1.01457533) which in turn equals \$116.60.

The answers for the simple and compound interest are the same because there is only one compounding period.

2.  To calculate APR use the formula APR = (interest paid / amount borrowed) x (365 / # of days of the loan).  In this case the interest paid is \$30, which is the amount repaid \$600 minus the amount borrowed \$570.  We dont know the exact number of days but we do know the number of months so we can adjust the last half of the formula to (12 / # of months of the loan).  We now have APR = (30 / 570) x (12/ / 3) so APR = .05263158 x 4 = .21052632 or approximately 21.05%.

The effective rate of interest he was charged is equal to ((1+interest paid / amount borrowed)365 / # of days of the loan) 1.  So ((1+.0563158)4 1) which becomes 1.24501649 1 or .24501649 or approximately 24.5%.

3.  This is solved using the compound interest (future value) equation FV = P x (1+i)n.  FV = \$5,000 x (1.085)1.5 so FV = \$5,000 x 1.13017217 = \$5,650.86.

4. This one is similar to number 2 above except in this case it is as an investor rather than as a borrower.  To get the APR we have ((9,890  9,780) / 9,780) x (365 / 46) which is (110 / 9,780) x 7.93478261 or .01124744 x 7.93478261 = .08924599 of 8.92+%.

5. When interest is quoted as being compounded on less than a yearly basis the quoted rate is equal to the sum of the effective periodic rates.  So, 8% compounded semi-annually is telling you that it pays 4% every six months.  That equates to and effective semi-annual rate of 4%.  The 4% effective semi-annual rate is converted to an effective annual rate by taking (1 + i)n 1;therefore we have (1.04)2 1 = .0816.  To get an effective daily rate you then take (1.0816)1/365-1 = (1.0816).00273973 1 = .00021087 or .021087% per day.